Hydraulic control, turbulence and mixing in stratified buoyancy-driven exchange flows

Buoyancy-driven exchange flows in geophysical contexts, such as flows through straits, often create a partially-mixed intermediate layer through mixing between the two stratified counterflowing turbulent layers. We present a three-layer hydraulic analysis of such flows, highlighting the dynamical importance of the intermediate layer. Our model is based on the viscous, shallow water, Boussinesq equations and includes the effects of mixing as a non-hydrostatic pressure forcing. We apply this shallow-water formulation to direct numerical simulations of stratified inclined duct (SID) exchange flows where turbulence is controlled by a modest slope of the duct. We show that the nonlinear characteristics of the three-layer model correspond to linear long waves perturbing the three-layer mean flow, and predict, in agreement with recent experimental observations in SID, hydraulically-controlled regions in the middle of the duct, linked to the onset of instability and turbulence. We also provide the first evidence of long-wave resonance, as well as resonance between long and short waves, and their connection to transitions from intermittent to fully developed turbulence. These results challenge current parameterisations for turbulent transport in stratified exchange flows, which typically overlook long waves and internal hydraulics induced by streamwise variations of the flow.